4-4 Rev. B. Bronwin on Jacobi's (general Transformation 



The third form, therefore, reduces to the first.^^ <^ "^^ sAmM^ 

 Also, __ ■■ ""''' '"-^"'^ 



sin^ am (2 w) = sin^ am (2 w — 2 K' \/ — 1) = si'n^ awi (2 w'), 

 if c«'=«;-K V'-l. Let ...i^ii T ^^ji^^vs.ae 



_ (2r + l) K + (2r'^r) ^t*^^^~^}^^ 



-!■ _ (2r+l)K + 2yVK^\/-l .-Ja.. 



which reduces the fourth form to the first. For, from what 



has been demonstrated, it follows that 



sin^ am {2mu)) =y{sin^ atn (2w)} =y{sin^ am {2u)')} 



= sin^ «7» {2mM') sin^co am (^mw) =:f^ {sin^ ain (2 a;)} 



= /i {sin^ am (2 co') } = sin^co atn (2 m co'), 

 where 7n is any integer, / and f denote certain functions, co 

 stands for one of the last three forms of this quantity, and w' 

 for the first form. To these we may add 

 cos^ am (2 m w) = 1 — sin^ am (2 w w) = 1 — sin^ «;« (2 m vJj 



= cos^ am (2 7W w'). 

 Jacobi's transformation reduced is 



«/ 



di/ 



or 



_ J^ /^-^ dx 



^ = iv? s^a{2(o) 



M 



s^ a (-i w) 



■-'id J.?;) l-^^^'''s2«(2a;)*l-FA-2s2„(4^) 



l- 



,>^^4f4' s2a(w— l)a; 



M 



1 — k^xh'^a {il — 1 ) cy J 



{sin CO am (2K;)sincoaw (4<ctf)... sin coam[n 

 si 



(«.) 



(1.) 





sin am (2cy) sin a?rt (4 w) ... sin am (w — 1) 

 A = ^" {sin CO am(2«;) sin CO am (4 w)... sin CO am(w—l)w}'*, 



where w may have any of the four forms. Let w again denote 

 any of the last three forms, co' the first. In (1.) and in the 

 expressions of M and A, we may change ca into w', and the 

 values of all the coefficients and all the constants will remain 

 unchanged. There is then but one transformation. 

 We easily transform (1.) into 



sa. Msa {u->r2oJ) sa(M4-(2« — 2) w') 



2J 



sa. w'sa(3w') sa (2w— l)w' 



(2.) 



